3.565 \(\int x (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{12} (x+1)^{12} (d-2 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{13} e (x+1)^{13} \]

[Out]

-((d - e)*(1 + x)^11)/11 + ((d - 2*e)*(1 + x)^12)/12 + (e*(1 + x)^13)/13

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Rubi [A]  time = 0.0353601, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {27, 76} \[ \frac{1}{12} (x+1)^{12} (d-2 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{13} e (x+1)^{13} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

-((d - e)*(1 + x)^11)/11 + ((d - 2*e)*(1 + x)^12)/12 + (e*(1 + x)^13)/13

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(d-2 e) (1+x)^{11}+e (1+x)^{12}\right ) \, dx\\ &=-\frac{1}{11} (d-e) (1+x)^{11}+\frac{1}{12} (d-2 e) (1+x)^{12}+\frac{1}{13} e (1+x)^{13}\\ \end{align*}

Mathematica [B]  time = 0.0154999, size = 147, normalized size = 3.77 \[ \frac{1}{12} x^{12} (d+10 e)+\frac{5}{11} x^{11} (2 d+9 e)+\frac{3}{2} x^{10} (3 d+8 e)+\frac{10}{3} x^9 (4 d+7 e)+\frac{21}{4} x^8 (5 d+6 e)+6 x^7 (6 d+5 e)+5 x^6 (7 d+4 e)+3 x^5 (8 d+3 e)+\frac{5}{4} x^4 (9 d+2 e)+\frac{1}{3} x^3 (10 d+e)+\frac{d x^2}{2}+\frac{e x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^2)/2 + ((10*d + e)*x^3)/3 + (5*(9*d + 2*e)*x^4)/4 + 3*(8*d + 3*e)*x^5 + 5*(7*d + 4*e)*x^6 + 6*(6*d + 5*e)
*x^7 + (21*(5*d + 6*e)*x^8)/4 + (10*(4*d + 7*e)*x^9)/3 + (3*(3*d + 8*e)*x^10)/2 + (5*(2*d + 9*e)*x^11)/11 + ((
d + 10*e)*x^12)/12 + (e*x^13)/13

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Maple [B]  time = 0., size = 130, normalized size = 3.3 \begin{align*}{\frac{e{x}^{13}}{13}}+{\frac{ \left ( d+10\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,d+e \right ){x}^{3}}{3}}+{\frac{d{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/13*e*x^13+1/12*(d+10*e)*x^12+1/11*(10*d+45*e)*x^11+1/10*(45*d+120*e)*x^10+1/9*(120*d+210*e)*x^9+1/8*(210*d+2
52*e)*x^8+1/7*(252*d+210*e)*x^7+1/6*(210*d+120*e)*x^6+1/5*(120*d+45*e)*x^5+1/4*(45*d+10*e)*x^4+1/3*(10*d+e)*x^
3+1/2*d*x^2

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Maxima [B]  time = 0.964514, size = 174, normalized size = 4.46 \begin{align*} \frac{1}{13} \, e x^{13} + \frac{1}{12} \,{\left (d + 10 \, e\right )} x^{12} + \frac{5}{11} \,{\left (2 \, d + 9 \, e\right )} x^{11} + \frac{3}{2} \,{\left (3 \, d + 8 \, e\right )} x^{10} + \frac{10}{3} \,{\left (4 \, d + 7 \, e\right )} x^{9} + \frac{21}{4} \,{\left (5 \, d + 6 \, e\right )} x^{8} + 6 \,{\left (6 \, d + 5 \, e\right )} x^{7} + 5 \,{\left (7 \, d + 4 \, e\right )} x^{6} + 3 \,{\left (8 \, d + 3 \, e\right )} x^{5} + \frac{5}{4} \,{\left (9 \, d + 2 \, e\right )} x^{4} + \frac{1}{3} \,{\left (10 \, d + e\right )} x^{3} + \frac{1}{2} \, d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/13*e*x^13 + 1/12*(d + 10*e)*x^12 + 5/11*(2*d + 9*e)*x^11 + 3/2*(3*d + 8*e)*x^10 + 10/3*(4*d + 7*e)*x^9 + 21/
4*(5*d + 6*e)*x^8 + 6*(6*d + 5*e)*x^7 + 5*(7*d + 4*e)*x^6 + 3*(8*d + 3*e)*x^5 + 5/4*(9*d + 2*e)*x^4 + 1/3*(10*
d + e)*x^3 + 1/2*d*x^2

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Fricas [B]  time = 1.3739, size = 371, normalized size = 9.51 \begin{align*} \frac{1}{13} x^{13} e + \frac{5}{6} x^{12} e + \frac{1}{12} x^{12} d + \frac{45}{11} x^{11} e + \frac{10}{11} x^{11} d + 12 x^{10} e + \frac{9}{2} x^{10} d + \frac{70}{3} x^{9} e + \frac{40}{3} x^{9} d + \frac{63}{2} x^{8} e + \frac{105}{4} x^{8} d + 30 x^{7} e + 36 x^{7} d + 20 x^{6} e + 35 x^{6} d + 9 x^{5} e + 24 x^{5} d + \frac{5}{2} x^{4} e + \frac{45}{4} x^{4} d + \frac{1}{3} x^{3} e + \frac{10}{3} x^{3} d + \frac{1}{2} x^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/13*x^13*e + 5/6*x^12*e + 1/12*x^12*d + 45/11*x^11*e + 10/11*x^11*d + 12*x^10*e + 9/2*x^10*d + 70/3*x^9*e + 4
0/3*x^9*d + 63/2*x^8*e + 105/4*x^8*d + 30*x^7*e + 36*x^7*d + 20*x^6*e + 35*x^6*d + 9*x^5*e + 24*x^5*d + 5/2*x^
4*e + 45/4*x^4*d + 1/3*x^3*e + 10/3*x^3*d + 1/2*x^2*d

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Sympy [B]  time = 0.175674, size = 133, normalized size = 3.41 \begin{align*} \frac{d x^{2}}{2} + \frac{e x^{13}}{13} + x^{12} \left (\frac{d}{12} + \frac{5 e}{6}\right ) + x^{11} \left (\frac{10 d}{11} + \frac{45 e}{11}\right ) + x^{10} \left (\frac{9 d}{2} + 12 e\right ) + x^{9} \left (\frac{40 d}{3} + \frac{70 e}{3}\right ) + x^{8} \left (\frac{105 d}{4} + \frac{63 e}{2}\right ) + x^{7} \left (36 d + 30 e\right ) + x^{6} \left (35 d + 20 e\right ) + x^{5} \left (24 d + 9 e\right ) + x^{4} \left (\frac{45 d}{4} + \frac{5 e}{2}\right ) + x^{3} \left (\frac{10 d}{3} + \frac{e}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**2/2 + e*x**13/13 + x**12*(d/12 + 5*e/6) + x**11*(10*d/11 + 45*e/11) + x**10*(9*d/2 + 12*e) + x**9*(40*d/3
+ 70*e/3) + x**8*(105*d/4 + 63*e/2) + x**7*(36*d + 30*e) + x**6*(35*d + 20*e) + x**5*(24*d + 9*e) + x**4*(45*
d/4 + 5*e/2) + x**3*(10*d/3 + e/3)

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Giac [B]  time = 1.13835, size = 194, normalized size = 4.97 \begin{align*} \frac{1}{13} \, x^{13} e + \frac{1}{12} \, d x^{12} + \frac{5}{6} \, x^{12} e + \frac{10}{11} \, d x^{11} + \frac{45}{11} \, x^{11} e + \frac{9}{2} \, d x^{10} + 12 \, x^{10} e + \frac{40}{3} \, d x^{9} + \frac{70}{3} \, x^{9} e + \frac{105}{4} \, d x^{8} + \frac{63}{2} \, x^{8} e + 36 \, d x^{7} + 30 \, x^{7} e + 35 \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 9 \, x^{5} e + \frac{45}{4} \, d x^{4} + \frac{5}{2} \, x^{4} e + \frac{10}{3} \, d x^{3} + \frac{1}{3} \, x^{3} e + \frac{1}{2} \, d x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/13*x^13*e + 1/12*d*x^12 + 5/6*x^12*e + 10/11*d*x^11 + 45/11*x^11*e + 9/2*d*x^10 + 12*x^10*e + 40/3*d*x^9 + 7
0/3*x^9*e + 105/4*d*x^8 + 63/2*x^8*e + 36*d*x^7 + 30*x^7*e + 35*d*x^6 + 20*x^6*e + 24*d*x^5 + 9*x^5*e + 45/4*d
*x^4 + 5/2*x^4*e + 10/3*d*x^3 + 1/3*x^3*e + 1/2*d*x^2